Walsh Cyclopropane Molecular Orbitals
Rigorous Derivations
Rigorous derivation of the A_{1}' cyclopropane MO from the A_{1} MOs of the methylene fragments.
There is one identical σtype orbital on each of the three
carbon atoms. Let the orbital on C_{1} be φ_{1s}, that on C_{2} be φ_{2s}, and that on C_{3} be
φ_{3s}.
We now apply the A_{1}' projection operator
P^{A1'} to one of the three orbitals, let us say
φ_{1s}. In order to do this, we
apply each symmetry operation in turn, multiplying the result by the character of the
A_{1}' representation.
D_{3h}  
E 
2C_{3} 
3C_{2} 
σ_{h} 
2S_{3} 
3σ_{v} 

A_{1}'  
1 
1 
1 
1 
1 
1 
P^{A1'}φ_{1s} 
≈ 
φ_{1s} +
φ_{2s} + φ_{3s} + φ_{1s} + φ_{2s} + φ_{3s} + φ_{1s} + φ_{2s} + φ_{3s} + φ_{2s} + φ_{3s} + φ_{1s} 

= 
4(φ_{1s} +
φ_{2s} + φ_{3s}) 

≈ 
φ_{1s} +
φ_{2s} + φ_{3s} 

The MO is therefore 


Normalized: 

ψ_{A1'} 
= 
1 
(φ_{1s} + φ_{2s} + φ_{3s}) 

√3 

Return
Rigorous derivation of the σtype E'
cyclopropane MOs from the A_{1} MOs of the methylene
fragments.
There is one identical σtype orbital on each of the three
carbon atoms. Let the orbital on C_{1} be φ_{1s}, that on C_{2} be φ_{2s}, and that on C_{3} be
φ_{3s}.
We now apply the E' projection operator P^{E'} to one of the three orbitals, let us say φ_{1s}. In order to do this, we apply
each symmetry operation in turn, multiplying the result by the character of the
E' representation.
D_{3h}  
E 
2C_{3} 
3C_{2} 
σ_{h} 
2S_{3} 
3σ_{v} 

E'  
2 
1 
0 
2 
1 
0 
P^{E'}φ_{1s} 
≈ 
2φ_{1s} 
φ_{2s}  φ_{3s} + 2φ_{1s}  φ_{2s}  φ_{3s} 

= 
2(2φ_{1s} 
φ_{2s}  φ_{3s}) 

≈ 
2φ_{1s} 
φ_{2s}  φ_{3s} 

The MO is therefore 


Normalized: 

ψ_{E',a} 
= 
1 
(2φ_{1s}  φ_{2s}  φ_{3s}) 

√6 

But we need another orbital. In order to do this, we need to pick a symmetry operation
belonging to the D_{3h} group which will convert ψ_{E',a} into something other
than ±1 times itself. For example, C_{3}:
C_{3}[ 
1 
(2φ_{1s}  φ_{2s}  φ_{3s}) 

√6 

] = 
1 
(2φ_{2s}  φ_{3s}  φ_{1s}) 

√6 

The new function is not orthogonal to ψ_{E',a}, and so it must be a linear combination of ψ_{E',a} with some other
function, ψ_{E',b}. To
find ψ_{E',b} we multiply
the new function by some appropriate factor, then subtract ψ_{E',a} from it:

2(2φ_{2s} 
φ_{3s}  φ_{1s}) + (2φ_{1s}  φ_{2s}  φ_{3s}) 
= 
4φ_{2s} 
2φ_{3s}  2φ_{1s} + 2φ_{1s}  φ_{2s}  φ_{3s} 
= 
3φ_{2s} 
3φ_{3s} ≈ φ_{2s}
 φ_{3s} 

The MO is
therefore 


Normalized: 

ψ_{E',b} 
= 
1 
(φ_{2s}  φ_{3s}) 

√2 

Return
Rigorous derivation of the A_{2}' cyclopropane MO from the B_{2} MOs of the methylene fragments.
There is one identical πtype orbital on each of the three
carbon atoms. Let the orbital on C_{1} be φ_{1p}, that on C_{2} be φ_{2p}, and that on C_{3} be
φ_{3p}.
We now apply the A_{2}' projection operator
P^{A2'} to one of the three orbitals, let us say
φ_{1p}. In order to do this, we
apply each symmetry operation in turn, multiplying the result by the character of the
A_{2}' representation.
D_{3h}  
E 
2C_{3} 
3C_{2} 
σ_{h} 
2S_{3} 
3σ_{v} 

A_{2}'  
1 
1 
1 
1 
1 
1 
P^{A2'}φ_{1p} 
≈ 
φ_{1p} +
φ_{2p} + φ_{3p}  (φ_{1p}  φ_{2p}  φ_{3p}) + φ_{1p} + φ_{2p} + φ_{3p}  (φ_{1p}  φ_{2p}  φ_{3p}) 

= 
4(φ_{1p} +
φ_{2p} + φ_{3p}) 

≈ 
φ_{1p} +
φ_{2p} + φ_{3p} 

The MO is therefore 


Normalized: 

ψ_{A2'} 
= 
1 
(φ_{1p} + φ_{2p} + φ_{3p}) 

√3 

Return
Rigorous derivation of the πtype E'
cyclopropane MOs from the B_{2} MOs of the methylene
fragments.
There is one identical πtype orbital on each of the three
carbon atoms. Let the orbital on C_{1} be φ_{1p}, that on C_{2} be φ_{2p}, and that on C_{3} be
φ_{3p}.
We now apply the E' projection operator P^{E'} to one of the three orbitals, let us say φ_{1p}. In order to do this, we apply
each symmetry operation in turn, multiplying the result by the character of the
E' representation.
D_{3h}  
E 
2C_{3} 
3C_{2} 
σ_{h} 
2S_{3} 
3σ_{v} 

E'  
2 
1 
0 
2 
1 
0 
P^{E'}φ_{1p} 
≈ 
2φ_{1p} 
φ_{2p}  φ_{3p} + 2φ_{1p}  φ_{2p}  φ_{3p} 

= 
2(2φ_{1p} 
φ_{2p}  φ_{3p}) 

≈ 
2φ_{1p} 
φ_{2p}  φ_{3p} 

The MO is therefore 


Normalized: 

ψ_{E',c} 
= 
1 
(2φ_{1p}  φ_{2p}  φ_{3p}) 

√6 

But we need another orbital. In order to do this, we need to pick a symmetry operation
belonging to the D_{3h} group which will convert ψ_{E',c} into something other
than ±1 times itself. For example, C_{3}:
C_{3}[ 
1 
(2φ_{1p}  φ_{2p}  φ_{3p}) 

√6 

] = 
1 
(2φ_{2p}  φ_{3p}  φ_{1p}) 

√6 

The new function is not orthogonal to ψ_{E',c}, and so it must be a linear combination of ψ_{E',c} with some other
function, ψ_{E',d}. To
find ψ_{E',d} we multiply
the new function by some appropriate factor, then subtract ψ_{E',c} from it:

2(2φ_{2p} 
φ_{3p}  φ_{1p}) + (2φ_{1p}  φ_{2p}  φ_{3p}) 
= 
4φ_{2p} 
2φ_{3p}  2φ_{1p} + 2φ_{1p}  φ_{2p}  φ_{3p} 
= 
3φ_{2p} 
3φ_{3p} ≈ φ_{2p}
 φ_{3p} 

The MO is therefore 


Normalized: 

ψ_{E',d} 
= 
1 
(φ_{2p}  φ_{3p}) 

√2 

Return
Copyright © 1997 by Daniel J. Berger. This work may be copied without limit if
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